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Full text of "History Of The Theory Of Numbers - I"

CHAP, ill]           GENERALIZATIONS OF FERMAT'° rpw* -----
Ed. Weyr164, E. Lucas165, and Pellet165 o divisible by N for any integers a, N.
H. Picquet166 noted the divisibility c tion of certain curvilinear polygons of _ and circumscribed in a given cubic curvv bility of F(a, N) by N, requiring variouf function F(a, N) is characterized by the t
(2)       F(a, np8) = F(ap°, n) -F(a?~\ rz>
where a is any integer, n an integer no-1 A. Grandi167 proved that F(a, N) ii
Each of these bhiomials is divisible by pi1 since
aCP-i)P-1asl>         a'W1
G. Koenigs168 considered a uniform si; power / =<l>n(z).   Those roots of z—cf)n(z) - „
of lower index are said to belong to the index ,«,. Ai u, ueiuugs to ine uiaex n, so do also &(&) for i=l,. . ., n-1. Thus the roots belonging to the index n are distributed into sets of n. If a is the degree of the polynomials in the numerator and denominator of <t>(z), the number of roots belonging to the index n is F(at n), which is therefore divisible by n.
MacMahon's112 paper contains in a disguised form the fact that F(a, N) is divisible by N. Proofs were given by E. Maillet113 by substitution groups, and by G. Cordone.169
Borel and Drach170 made use of Gauss' result that F(p} N) is divisible by N for every prime p and integer N, and Dirichlet's theorem that there exist an infinitude of primes p congruent modulo N to any given integer a prime to N, to conclude that F(a, N) is divisible by N.
L. E. Dickson171 proved by induction (from k to k+l prunes) that F(a, N) is characterized by properties (2) and concluded by induction that F(a, N) is divisible by N. A like conclusion was drawn from
\F(a, N)\'-F(a, N)^F(a, qN) (mod <?), where q is a prune.    He gave the relations
F(a, nN) = F(aN, n) - 2 F(a"/w, n) + S F(aw/ww, n) - . . .
»*Casopis, Prag, 11, 1882, 39.
"'Comptes Rendus Paris, 96, 1883, 1300-2.
™Ibid., p. 1136, 1424.    Jour, dc I'Scole polyt., cah. 54, 1884, 61, 85-91.
"7Atti R. Istituto Vcncto di Sc., (6), 1, 1882-3, 809.
»«Bull. des sciences math., (2), 8, 1884, 286.
"•Rivista di Mat., Torino, 5, 1895, 25.
170Introd. th6orie des nombres, 1895, 50.
mAnnala of Math., (2), 1, 1899, 35.   Abatr. in Comptea Rendua Paris, 128, 1899, 1083-5.